Fuzzy Control Tutorial

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Fuzzy Control Tutorial. Dr. Stephen Paul Linder. What is fuzzy?. A dictionary definition Of or resembling fuzz. Not clear; indistinct: a fuzzy recollection of past events. Not coherent; confused: a fuzzy plan of action. Covered with fuzz. And so what is a Fuzzy Set? a not clear Set?.

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Fuzzy Control Tutorial

Dr. Stephen Paul Linder

What is fuzzy?
• A dictionary definition
• Of or resembling fuzz.
• Not clear; indistinct: a fuzzy recollection of past events.
• Not coherent; confused: a fuzzy plan of action.
• Covered with fuzz.
• And so what is a Fuzzy Set?
• a not clear Set?

Stephen Linder

Fuzzy Sets
• , "Fuzzy sets," Information and Control, vol. 8, pp. 338--353, 1965.
• A generalization of set theory that allows partial membership in a set.
• Membership is a real number with a range [0, 1]
• Membership functions are commonly triangular or Gaussian because ease of computation.
• Utility comes from overlapping membership functions – a value can belong to more than one set

Stephen Linder

Precise versus fuzzy statements
• Sally is tall
• Sally if 5’10”.
• If Sally is on the basket ball team: Sally is 6’4”.
• It is cold outside.
• In the winterIt is 12° F outside.
• In the summer:It is 60° F outside.
• In the summer in northern Canada It is 30° F outside.

Stephen Linder

Example Membership functions

(x = 0 )

Not stretching or compressing

Spring is compressing fast

Spring is stretching fast

small

Large

Small

1.0

1.0

Membership

0.0

0.0

Spring Stretching

Velocity

Large Negative

Zero

Large Positive

Zero

Negative

Positive

1.0

1.0

0.0

0.0

Position Error

Spring Length

Stephen Linder

A few rules can make complex decision surfaces

Control Output

Spring State

Velocity

Constructed from 10 rules

Stephen Linder

A few rules can make complex decision surfaces

Control Output

Position Error

Spring State

Constructed from 10 rules

Stephen Linder

Fuzzy vs. Probabilistic Reasoning
• Probabilistic Reasoning
• "There is an 80% chance that Jane is old"
• Jane is either old or not old (the law of the excluded middle).
• Fuzzy Reasoning
• "Jane's degree of membership within the set of old people is 0.80.”
• Jane is like an old person, but could also have some characteristics of a young person.

Stephen Linder

Why fuzzy?
• Precision is not truth.— Henri Matisse
• So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality. — Albert Einstein
• As complexity rises, precise statements lose meaning and meaningful statements lose precision. — Lotfi Zadeh

Stephen Linder

Why the reluctance use of fuzzy logic?
• Engineers are trained using precise mathematics – differential equations
• Most of us are more comfortable with the Law of the Excluded Middle
• every proposition must either be True or False
• The use of the word fuzzy.
• What if AI were call Epistemological Engineering as suggested in 1968 at the Machine Intelligence workshop in Edinburgh?
• Not enough software people are in charge of engineering projects

Stephen Linder

Fuzzy Control: Inverted Pendulum Problem

State variables

Angle of the Pendulum

Rate of change of the angle

Position of the cart

Problem

Keep pendulum upright by moving cart left or right.

Stephen Linder

Partition variables

Pendulum Angle

Inputs

Pendulum Angular

Velocity

Output

Cart Speed

Stephen Linder

Controller Rules

If angle is zero and angular velocity is zero

then speed shall be zero.

This is an example of a Fuzzy PD Controller!

Stephen Linder

Example input

Input is both zero and positive low.

Input is both zero and negative low.

How many rules will be fired?

Stephen Linder

Example output from one rule

if angle is zero and angular velocity is zero then

Stephen Linder

Fused output from four rules

if angle is zero and angular velocity is zero then

if angle is zero and angular velocity is negative low then

if angle is positive low and angular velocity is zero then

Defuzzification must now be done on fused output.

What are some possible defuzzification methods?

if angle is positive low and angular velocity is negative low then

Stephen Linder

A Java-based Simulation
• Fuzzy Pendulum Demo created using the FuzzyJ Toolkit by the Integrated Reasoning Group of the National Research Council of Canada
• http://www.iit.nrc.ca/IR_public/fuzzy/FuzzyPendulum.html

Stephen Linder

Conclusions
• Fuzzy Control allows someone to do control with a relatively small set of rules because
• fuzzy sets overlap
• the output of conflicting rules can be merge to create complex behaviors
• Fuzzy controllers are faster for engineers to design if they have no control experience
• Fuzzy control requires a wider range of design patterns than just fuzzy version of classical controls

Stephen Linder